All-order differential equations for one-loop closed-string integrals and modular graph forms

Abstract

We investigate generating functions for the integrals over world-sheet tori appearing in closed-string one-loop amplitudes of bosonic, heterotic and type-II theories. These closed-string integrals are shown to obey homogeneous and linear differential equations in the modular parameter of the torus. We spell out the first-order Cauchy-Riemann and secondorder Laplace equations for the generating functions for any number of external states. The low-energy expansion of such torus integrals introduces infinite families of non-holomorphic modular forms known as modular graph forms. Our results generate homogeneous first-and second-order differential equations for arbitrary such modular graph forms and can be viewed as a step towards all-order low-energy expansions of closed-string integrals. arXiv:1911.03476v2 [hep-th] 21 Jan 2020 F Derivation of component equations at three points 65 F.1 General Cauchy-Riemann component equations at three points 66 F.2 Examples of Cauchy-Riemann component equations at three points 67 F.3 Further examples for Laplace equations at three points 68 -ii -10The subscript of ∇DG aims to avoid confusion with the differential operators of Sections 3.1 and 3.2 and refers to the authors D'Hoker and Green of [9] which initiated the systematic study of relations between modular graph forms via repeated action of (2.59).

Generating series of all modular graph forms from iterated Eisenstein integrals

“…advantage of working with generating series is that their differential equations in τ , derived in our previous work [30], are valid to all orders in α and take a simple form for any number n of punctures.…”

From little string free energies towards modular graph functions

“…The coupling functions O (N ),i for i > 1 are somewhat more involved. However, they still share a fair amount of properties with so-called graph functions, which have appeared in the study of Feynman diagrams in field theory as well as string theory [32,33,39,40,38,[41][42][43][44][45]. To make contact with these recent results in the literature, we choose to represent (5.4) in a slightly different fashion [22] O (2),1 = −2 I 0 ,…”

“…[27][28][29][30][31][32][33][34][35][36][37][38] and references therein for an overview). The current paper is motivated by extending this connection and extracting a class of functions from the free energy F N,1 , which show a certain resemblance of so-called (modular) graph functions (see [32,33,39,40,38,[41][42][43][44][45]) that have been studied recently in the literature. To this end, we consider the so-called unrefined limit 1 = − 2 = and study instanton expansions of the free energy for N = 2, 3 up to order Q 3 R :…”

Transcendentality violation in type IIB string amplitudes

“…In fact, as mentioned before, unitarity cut construction of the one loop amplitude determines e k in terms of ζ sv (2n + 1) with n ≥ 1. 32 In fact, this is indeed the case upto the s 6 R 4 term [1].…”

“…Now while topologically there are several graphs at every order in the α ′ expansion, there are several algebraic relations between them which vastly reduce the number of them that have to be integrated individually over the moduli space of the complex structure of the torus. This feature, coupled with Poisson equations the modular graphs satisfy on the worldsheet moduli space and their asymptotic expansions around the cusp τ 2 → ∞, is helpful in performing these integrals [4,[24][25][26][27][28][29][30][31][32] and obtaining the coefficients of the various terms in the effective action. 24 We denote ij... Based on known results for the worldsheet analysis for the first few terms in the α ′ expansion upto the D 12 R 4 term, a natural pattern emerges for the final expression for the integrand over moduli space at arbitrary orders in the α ′ expansion 25 .…”